Select Source:

# Covariance

BIBLIOGRAPHY

The covariance is a measure of the magnitude of association between the scores of cases on two variables that have been measured at the interval or ratio level. It describes both the direction and the strength of the association. In

the social sciences, the covariance is most commonly used in structural equation modeling of systems of linear equations of measured and unmeasured variables.

Formally, the covariance between the scores of c cases (i through N ) on the variables X and Y is:

That is: subtract the first cases score on X from the mean of X; subtract the first cases score on Y from the mean of Y; multiply these deviations. Repeat this process for all of the cases, and sum the results. Divide this product by the population size (N).

When the relationship between X and Y is being examined in a random sample of cases drawn from the population, N -1 is usually substituted in the denominator. Most statistical software uses N -1.

The covariance of a variable with itself (e.g., COV (X, X)) is the variance. (For a more in-depth formal treatment of the covariance, see Snedecor and Cochran 1980).

If there is a tendency for higher scores on X to cooccur with higher scores on Y, the covariance will have a positive value; if there is a tendency for higher scores on X to co-occur with lower scores on Y, the covariance will be negative. If the scores on two variables are not associated, the covariance will equal zero. The units of measurement of the covariance are XY; for example, if X was measured in dollars, and Y was measured in years, the magnitude of the covariance would be dollar-years. When we are working with multiple variables, the variances and covariances among all the variables are arrayed in a symmetric variance-covariance matrix.

Consider the relationship shown in the scatter-plot, between the level of urbanization (X) and female life expectancy (Y) in nineteen African countries in the mid-1990s. Inspection suggests that the scores positively covary: on the average (but not in all cases), the higher the urbanization, the higher the life expectancy.

The covariance for this relationship is 57.538. The positive value indicates a positive relationship. The strength of the relationship is difficult to assess because the unit of measurement of the covariance is percent-years. Because of this peculiar metric, the covariance is rarely used as a simple description. The Pearson correlation (which is. 47 in this example) is preferred.

The covariance is the most commonly used measure of association when research involves predicting Y from X using structural equation modeling. In predictive modeling, there is often the desire to describe the relationship between Y and X in the original scales of the variables: How much Y do we get for each unit of X ?

Some warnings: Restricted variation in either variable, non-linearity in the relationship, and non-normality in the joint distribution of X and Y can limit the validity of the covariance as an index of the strength and direction of the relationship.

## BIBLIOGRAPHY

Snedecor, George W., and William Gemmell Cochran. 1980. Statistical Methods. 7th ed. Ames: Iowa State Press.

Robert Hanneman

Pick a style below, and copy the text for your bibliography.

• MLA
• Chicago
• APA

"Covariance." International Encyclopedia of the Social Sciences. . Encyclopedia.com. 23 Mar. 2018 <http://www.encyclopedia.com>.

"Covariance." International Encyclopedia of the Social Sciences. . Encyclopedia.com. (March 23, 2018). http://www.encyclopedia.com/social-sciences/applied-and-social-sciences-magazines/covariance

"Covariance." International Encyclopedia of the Social Sciences. . Retrieved March 23, 2018 from Encyclopedia.com: http://www.encyclopedia.com/social-sciences/applied-and-social-sciences-magazines/covariance

### Citation styles

Encyclopedia.com gives you the ability to cite reference entries and articles according to common styles from the Modern Language Association (MLA), The Chicago Manual of Style, and the American Psychological Association (APA).

Within the “Cite this article” tool, pick a style to see how all available information looks when formatted according to that style. Then, copy and paste the text into your bibliography or works cited list.

Because each style has its own formatting nuances that evolve over time and not all information is available for every reference entry or article, Encyclopedia.com cannot guarantee each citation it generates. Therefore, it’s best to use Encyclopedia.com citations as a starting point before checking the style against your school or publication’s requirements and the most-recent information available at these sites:

#### Modern Language Association

http://www.mla.org/style

#### The Chicago Manual of Style

http://www.chicagomanualofstyle.org/tools_citationguide.html

#### American Psychological Association

http://apastyle.apa.org/

##### Notes:
• Most online reference entries and articles do not have page numbers. Therefore, that information is unavailable for most Encyclopedia.com content. However, the date of retrieval is often important. Refer to each style’s convention regarding the best way to format page numbers and retrieval dates.
• In addition to the MLA, Chicago, and APA styles, your school, university, publication, or institution may have its own requirements for citations. Therefore, be sure to refer to those guidelines when editing your bibliography or works cited list.

## covariance

covariance A measure of the joint variation of two random variables, analogous to variance (see measures of variation). If the variables are x and y then the covariance of x and y is Σ(xi)(yi)

The analysis of covariance is an extension of the analysis of variance in which the variables to be tested are adjusted to take account of assumed linear relationships with other variables. See also correlation.

Pick a style below, and copy the text for your bibliography.

• MLA
• Chicago
• APA

"covariance." A Dictionary of Computing. . Encyclopedia.com. 23 Mar. 2018 <http://www.encyclopedia.com>.

"covariance." A Dictionary of Computing. . Encyclopedia.com. (March 23, 2018). http://www.encyclopedia.com/computing/dictionaries-thesauruses-pictures-and-press-releases/covariance

"covariance." A Dictionary of Computing. . Retrieved March 23, 2018 from Encyclopedia.com: http://www.encyclopedia.com/computing/dictionaries-thesauruses-pictures-and-press-releases/covariance

### Citation styles

Encyclopedia.com gives you the ability to cite reference entries and articles according to common styles from the Modern Language Association (MLA), The Chicago Manual of Style, and the American Psychological Association (APA).

Within the “Cite this article” tool, pick a style to see how all available information looks when formatted according to that style. Then, copy and paste the text into your bibliography or works cited list.

Because each style has its own formatting nuances that evolve over time and not all information is available for every reference entry or article, Encyclopedia.com cannot guarantee each citation it generates. Therefore, it’s best to use Encyclopedia.com citations as a starting point before checking the style against your school or publication’s requirements and the most-recent information available at these sites:

#### Modern Language Association

http://www.mla.org/style

#### The Chicago Manual of Style

http://www.chicagomanualofstyle.org/tools_citationguide.html

#### American Psychological Association

http://apastyle.apa.org/

##### Notes:
• Most online reference entries and articles do not have page numbers. Therefore, that information is unavailable for most Encyclopedia.com content. However, the date of retrieval is often important. Refer to each style’s convention regarding the best way to format page numbers and retrieval dates.
• In addition to the MLA, Chicago, and APA styles, your school, university, publication, or institution may have its own requirements for citations. Therefore, be sure to refer to those guidelines when editing your bibliography or works cited list.

## covariance

covariance In statistics, a measure of the association between two variables. Covariance is calculated as the difference between the average product of corresponding values in the two data sets and the product of the means of the two data sets. A value of zero indicates no relationship between the data sets. The covariance value can be used to calculate linear regression and principal components.

Pick a style below, and copy the text for your bibliography.

• MLA
• Chicago
• APA

"covariance." A Dictionary of Earth Sciences. . Encyclopedia.com. 23 Mar. 2018 <http://www.encyclopedia.com>.

"covariance." A Dictionary of Earth Sciences. . Encyclopedia.com. (March 23, 2018). http://www.encyclopedia.com/science/dictionaries-thesauruses-pictures-and-press-releases/covariance

"covariance." A Dictionary of Earth Sciences. . Retrieved March 23, 2018 from Encyclopedia.com: http://www.encyclopedia.com/science/dictionaries-thesauruses-pictures-and-press-releases/covariance

### Citation styles

Encyclopedia.com gives you the ability to cite reference entries and articles according to common styles from the Modern Language Association (MLA), The Chicago Manual of Style, and the American Psychological Association (APA).

Within the “Cite this article” tool, pick a style to see how all available information looks when formatted according to that style. Then, copy and paste the text into your bibliography or works cited list.

Because each style has its own formatting nuances that evolve over time and not all information is available for every reference entry or article, Encyclopedia.com cannot guarantee each citation it generates. Therefore, it’s best to use Encyclopedia.com citations as a starting point before checking the style against your school or publication’s requirements and the most-recent information available at these sites:

#### Modern Language Association

http://www.mla.org/style

#### The Chicago Manual of Style

http://www.chicagomanualofstyle.org/tools_citationguide.html

#### American Psychological Association

http://apastyle.apa.org/

##### Notes:
• Most online reference entries and articles do not have page numbers. Therefore, that information is unavailable for most Encyclopedia.com content. However, the date of retrieval is often important. Refer to each style’s convention regarding the best way to format page numbers and retrieval dates.
• In addition to the MLA, Chicago, and APA styles, your school, university, publication, or institution may have its own requirements for citations. Therefore, be sure to refer to those guidelines when editing your bibliography or works cited list.